%I #16 Oct 21 2024 08:59:38

%S 0,27,297,24570,267030,22064157,239792967,19813588740,215333817660,

%T 17792580624687,193369528466037,15977717587380510,173645621228683890,

%U 14347972600887073617,155933574493829667507,12884463417879004727880,140028176249837812737720

%N Values of n such that 2*n+1 and 7*n+1 are both triangular numbers (A000217).

%C Intersection of A074377 and A274830.

%H Colin Barker, <a href="/A274832/b274832.txt">Table of n, a(n) for n = 1..650</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,898,-898,-1,1).

%F G.f.: 27*x^2*(1+10*x+x^2) / ((1-x)*(1-30*x+x^2)*(1+30*x+x^2)).

%e 27 is in the sequence because 2*27+1 = 55, 7*27+1 = 190, and 55 and 190 are both triangular numbers.

%t LinearRecurrence[{1, 898, -898, -1, 1}, {0, 27, 297, 24570, 267030}, 20] (* _Paolo Xausa_, Oct 21 2024 *)

%o (PARI) isok(n) = ispolygonal(2*n+1, 3) && ispolygonal(7*n+1, 3)

%o (PARI) concat(0, Vec(27*x^2*(1+10*x+x^2)/((1-x)*(1-30*x+x^2)*(1+30*x+x^2)) + O(x^20)))

%Y Cf. A000217, A074377, A274830.

%Y Cf. A124174 (2*n+1 and 9*n+1), A274579 (2*n+1 and 5*n+1), A274603 (2*n+1 and 3*n+1), A274680 (2*n+1 and 4*n+1), A274756 (2*n+1 and 7*n+1).

%K nonn,easy

%O 1,2

%A _Colin Barker_, Jul 08 2016